How do I solve this:
The absolute value of ax+b is less than or equal to c for the terms of x in terms of a, b, and c where a>0 and c>0
-Then they ask:
The absolute value of ax+b is more than or equal to c for the terms of x in terms of a, b, and c where a>0 and c>0
It's the same thing except the second one is more than...
To solve the absolute value inequality |ax+b| ≤ c, where a > 0 and c > 0, you can follow these steps:
1. Set up two separate cases for the inequality:
Case 1: ax + b ≥ 0
Case 2: ax + b < 0
2. For Case 1, isolate the absolute value term:
ax + b ≤ c
3. Solve for x by subtracting b from both sides:
ax ≤ c - b
4. Divide both sides by a:
x ≤ (c - b)/a
This is the solution for Case 1.
5. For Case 2, isolate the absolute value term:
-(ax + b) ≤ c
6. Solve for x by multiplying both sides by -1 and flipping the inequality sign:
ax + b ≥ -c
7. Subtract b from both sides:
ax ≥ -c - b
8. Divide both sides by a:
x ≥ (-c - b)/a
This is the solution for Case 2.
To solve the absolute value inequality |ax+b| ≥ c, where a > 0 and c > 0, you can follow the same steps as above, but with a slight modification in Step 5:
5. For Case 2, isolate the absolute value term:
-(ax + b) ≥ c
6. Solve for x by multiplying both sides by -1 and flipping the inequality sign:
ax + b ≤ -c
7. Subtract b from both sides:
ax ≤ -c - b
8. Divide both sides by a:
x ≤ (-c - b)/a
This is the solution for Case 2.
The key difference in the second question is that the inequality sign is flipped in Step 6 for Case 2, resulting in different solutions for the two cases.